As is not terribly uncommon when it comes to Dr Kaku's media appearances on matters other than science, there do appear to be some flaws in his comment.

For example, the Paquisha War between Ecuador and Peru in 1981, though not exactly on par with WWII, was fought between two relatively new democracies.

1969's Football War between El Salvador and Honduras, other than being the last appearance of piston-engined fighters in wartime against each other (late variants of the Corsair and Mustangs), was between democracies.

The breakup of Yugoslavia, beginning in 1991, saw Croatia and Slovenia declare independence from the greater country, later known as Serbia. All three of these belligerents had democratically elected leadership at the time.

Going further back, you could count The Boer Wars and the Mexican-American War as well.

Of course, with a suitably strict set of guidelines as to what a democracy is, you can eliminate all of these.